Applying the Rules of Multiplication

Oct 13, 2025 by wbrum.org 143 views

Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the rules of multiplication and how to simplify complex expressions. In this case, we're given the expression $\frac{5k}{6} \cdot \frac{3}{2k^3}$ and asked to find its product. To solve this problem, we'll apply the rules of multiplication and simplify the resulting expression.

Applying the Rules of Multiplication

To multiply two fractions, we multiply the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately. In this case, we have:

$\frac{5k}{6} \cdot \frac{3}{2k^3}$

To multiply the numerators, we multiply $5k$ and $3$ , which gives us $15k$ . To multiply the denominators, we multiply $6$ and $2k^3$ , which gives us $12k^3$ .

Simplifying the Expression

Now that we have the product of the numerators and denominators, we can simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF). In this case, the GCF of $15k$ and $12k^3$ is $3k$ . Dividing both the numerator and denominator by $3k$ gives us:

$\frac{15k}{12k^3} \div \frac{3k}{1}$

Canceling Out Common Factors

When we divide two fractions, we can cancel out common factors in the numerator and denominator. In this case, we can cancel out the $3k$ in the numerator and denominator, which gives us:

$\frac{5}{4k^2}$

Conclusion

Therefore, the product of the given expression $\frac{5k}{6} \cdot \frac{3}{2k^3}$ is $\frac{5}{4k^2}$ .

Answer Key

The correct answer is A. $\frac{5}{4k^2}$ .

Tips and Tricks

When dealing with algebraic expressions, it's essential to understand the rules of multiplication and how to simplify complex expressions. Here are some tips and tricks to help you solve similar problems:

Always multiply the numerators and denominators separately.
Simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF).
Cancel out common factors in the numerator and denominator when dividing fractions.
Practice, practice, practice! The more you practice, the more comfortable you'll become with simplifying complex expressions.

Common Mistakes to Avoid

When dealing with algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Not multiplying the numerators and denominators separately.
Not simplifying the expression by dividing both the numerator and denominator by their greatest common factor (GCF).
Not canceling out common factors in the numerator and denominator when dividing fractions.
Not practicing enough! The more you practice, the more comfortable you'll become with simplifying complex expressions.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Here are a few examples:

In physics, algebraic expressions are used to describe the motion of objects. Simplifying these expressions can help physicists understand the behavior of complex systems.
In engineering, algebraic expressions are used to design and optimize systems. Simplifying these expressions can help engineers create more efficient and effective systems.
In finance, algebraic expressions are used to model and analyze financial data. Simplifying these expressions can help financial analysts make more informed decisions.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill that has many real-world applications. By understanding the rules of multiplication and how to simplify complex expressions, you can solve a wide range of problems and make more informed decisions. Remember to always multiply the numerators and denominators separately, simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF), and cancel out common factors in the numerator and denominator when dividing fractions. With practice and patience, you'll become a master of simplifying algebraic expressions in no time!

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to multiply the numerators and denominators separately.

Q: How do I multiply the numerators and denominators?

A: To multiply the numerators and denominators, you multiply the numbers on top (the numerators) and the numbers on the bottom (the denominators) separately.

Q: What is the next step in simplifying an algebraic expression?

A: The next step in simplifying an algebraic expression is to simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF).

Q: How do I find the greatest common factor (GCF) of two numbers?

A: To find the greatest common factor (GCF) of two numbers, you need to list the factors of each number and find the greatest factor they have in common.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to cancel out common factors in the numerator and denominator when dividing fractions.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to make the expression easier to understand and work with. It also helps to eliminate unnecessary complexity and make the expression more manageable.

Q: Can you give an example of how to simplify an algebraic expression?

A: Let's say we have the expression $\frac{12x}{18y}$ . To simplify this expression, we would multiply the numerators and denominators separately, which gives us $\frac{12x \cdot 1}{18y \cdot 1}$ . Then, we would simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF), which is 6. This gives us $\frac{2x}{3y}$ .

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include not multiplying the numerators and denominators separately, not simplifying the expression by dividing both the numerator and denominator by their greatest common factor (GCF), and not canceling out common factors in the numerator and denominator when dividing fractions.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through a series of problems and exercises. You can also try simplifying expressions on your own and then checking your work to make sure you got the correct answer.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including physics, engineering, and finance. In physics, algebraic expressions are used to describe the motion of objects, and simplifying these expressions can help physicists understand the behavior of complex systems. In engineering, algebraic expressions are used to design and optimize systems, and simplifying these expressions can help engineers create more efficient and effective systems. In finance, algebraic expressions are used to model and analyze financial data, and simplifying these expressions can help financial analysts make more informed decisions.

Q: Can you give some tips for simplifying algebraic expressions?

A: Here are some tips for simplifying algebraic expressions:

Always multiply the numerators and denominators separately.
Simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF).
Cancel out common factors in the numerator and denominator when dividing fractions.
Practice, practice, practice! The more you practice, the more comfortable you'll become with simplifying complex expressions.

Q: What are some common algebraic expressions that can be simplified?

A: Some common algebraic expressions that can be simplified include fractions, decimals, and percentages. You can also simplify expressions with variables, such as x and y.

Q: Can you give an example of how to simplify a fraction?

A: Let's say we have the fraction $\frac{12}{18}$ . To simplify this fraction, we would divide both the numerator and denominator by their greatest common factor (GCF), which is 6. This gives us $\frac{2}{3}$ .

Q: Can you give an example of how to simplify a decimal?

A: Let's say we have the decimal $0.12$ . To simplify this decimal, we would convert it to a fraction by writing it as $\frac{12}{100}$ . Then, we would simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 4. This gives us $\frac{3}{25}$ .

Q: Can you give an example of how to simplify a percentage?

A: Let's say we have the percentage $25\%$ . To simplify this percentage, we would convert it to a decimal by dividing it by 100. This gives us $0.25$ . Then, we would convert the decimal to a fraction by writing it as $\frac{25}{100}$ . Finally, we would simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 25. This gives us $\frac{1}{4}$ .

Q: Can you give an example of how to simplify an expression with variables?

Q: Can you give an example of how to simplify an expression with exponents?

A: Let's say we have the expression $\frac{2^3}{2^2}$ . To simplify this expression, we would use the rule that $a^m \cdot a^n = a^{m+n}$ . This gives us $\frac{2^{3+2}}{2^2} = \frac{2^5}{2^2}$ . Then, we would simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF), which is $2^2$ . This gives us $2^{5-2} = 2^3$ .

Q: Can you give an example of how to simplify an expression with fractions and exponents?

A: Let's say we have the expression $\frac{2^3}{2^2} \cdot \frac{3^2}{3^1}$ . To simplify this expression, we would use the rule that $a^m \cdot a^n = a^{m+n}$ . This gives us $\frac{2^{3+2}}{2^2} \cdot \frac{3^{2+1}}{3^1} = \frac{2^5}{2^2} \cdot \frac{3^3}{3^1}$ . Then, we would simplify the expression by dividing both the numerator and denominator by their greatest common factor (GCF), which is $2^2$ and $3^1$ . This gives us $2^{5-2} \cdot 3^{3-1} = 2^3 \cdot 3^2$ .

Q: Can you give an example of how to simplify an expression with decimals and exponents?

A: Let's say we have the expression $0.12 \cdot 2^3$ . To simplify this expression, we would convert the decimal to a fraction by writing it as $\frac{12}{100}$ . Then, we would simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 4. This gives us $\frac{3}{25}$ . Next, we would use the rule that $a^m \cdot a^n = a^{m+n}$ . This gives us $\frac{3}{25} \cdot 2^{3+1} = \frac{3}{25} \cdot 2^4$ . Then, we would simplify the expression by multiplying the numerator and denominator by $2^4$ . This gives us $\frac{3 \cdot 2^4}{25 \cdot 2^4} = \frac{3 \cdot 16}{25 \cdot 16} = \frac{48}{400}$ . Finally, we would simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 8. This gives us $\frac{6}{50}$ .

Q: Can you give an example of how to simplify an expression with percentages and exponents?

A: Let's say we have the expression $25\% \cdot 2^3$ . To simplify this expression, we would convert the percentage to a decimal by dividing it by 100. This gives us $0.25$ . Then, we would convert the decimal to a fraction by writing it as $\frac{25}{100}$ . Next, we would simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 25. This gives us $\frac{1}{4}$ . Next, we would use the rule that $a^m \cdot a^n = a^{m+n}$ . This gives us $\frac{1}{4} \cdot 2^{3+1} = \frac{1}{4} \cdot 2^4$ . Then, we would simplify the expression by multiplying the numerator and denominator by $2^4$ . This gives us $\frac{1 \cdot 16}{4 \cdot